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1. Bifurcations of an elastic disc coated with an elastic inextensible rod

   https://doi.org/10.1098/rspa.2023.0491  

   Gaibotti, M; Mogilevskaya, S G; Piccolroaz, A; Bigoni, D (January 2024, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   An analytical solution is derived for the bifurcations of an elastic disc that is constrained on the boundary with an isoperimetric Cosserat coating. The latter is treated as an elastic circular rod, either perfectly or partially bonded (with a slip interface in the latter case) and is subjected to three different types of uniformly distributed radial loads (including hydrostatic pressure). The proposed solution technique employs complex potentials to treat the disc’s interior and incremental Lagrangian equations to describe the prestressed elastic rod modelling the coating. The bifurcations of the disc occur with modes characterized by different circumferential wavenumbers, ranging between ovalization and high-order waviness, as a function of the ratio between the elastic stiffness of the disc and the bending stiffness of its coating. The presented results find applications in various fields, such as coated fibres, mechanical rollers, and the growth and morphogenesis of plants and fruits. 

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2. Elastic solids with strain-gradient elastic boundary surfaces

   https://doi.org/10.1007/s10659-024-10073-w  

   Rodriguez, C (June 2024, Journal of Elasticity)

   Recent works have shown that in contrast to classical linear elastic fracture mechanics, endowing crack fronts in a brittle Green-elastic solid with Steigmann-Ogden surface elasticity yields a model that predicts bounded stresses and strains at the crack tips for plane-strain problems. However, singularities persist for anti-plane shear (mode-III fracture) under far field loading, even when Steigmann-Ogden surface elasticity is incorporated. This work is motivated by obtaining a model of brittle fracture capable of predicting bounded stresses and strains for all modes of loading. We formulate an exact general theory of a three-dimensional solid containing a boundary surface with strain-gradient surface elasticity. For planar reference surfaces parameterized by flat coordinates, the form of surface elasticity reduces to that introduced by Hilgers and Pipkin, and when the surface energy is independent of the surface covariant derivative of the stretching, the theory reduces to that of Steigmann and Ogden. We discuss material symmetry using Murdoch and Cohen’s extension of Noll’s theory. We present a model small-strain surface energy that incorporates resistance to geodesic distortion, satisfies strong ellipticity, and requires the same material constants found in the Steigmann-Ogden theory. Finally, we derive and apply the linearized theory to mode-III fracture in an infinite plate under far-field loading. We prove that there always exists a unique classical solution to the governing integro-differential equation, and in contrast to using Steigmann-Ogden surface elasticity, our model is consistent with the linearization assumption in predicting finite stresses and strains at the crack tips. 

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3. Elastic Thermobarometry

   https://doi.org/10.1146/annurev-earth-031621-112720  

   Kohn, Matthew J.; Mazzucchelli, Mattia L.; Alvaro, Matteo (May 2023, Annual Review of Earth and Planetary Sciences)

   Upon exhumation and cooling, contrasting compressibilities and thermal expansivities induce differential strains (volume mismatches) between a host crystal and its inclusions. These strains can be quantified in situ using Raman spectroscopy or X-ray diffraction. Knowing equations of state and elastic properties of minerals, elastic thermobarometry inverts measured strains to calculate the pressure-temperature conditions under which the stress state was uniform in the host and inclusion. These are commonly interpreted to represent the conditions of inclusion entrapment. Modeling and experiments quantify corrections for inclusion shape, proximity to surfaces, and (most importantly) crystal-axis anisotropy, and they permit accurate application of the more common elastic thermobarometers. New research is exploring the conditions of crystal growth, reaction overstepping, and the magnitudes of differential stresses, as well as inelastic resetting of inclusion and host strain, and potential new thermobarometers for lower-symmetry minerals. ▪ A physics-based method is revolutionizing calculations of metamorphic pressures and temperatures. ▪ Inclusion shape, crystal anisotropy, and proximity to boundaries affect calculations but can be corrected for. ▪ New results are leading petrologists to reconsider pressure-temperature conditions, differential stresses, and thermodynamic equilibrium. 

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4. ELASTIC GRAPHS

   https://doi.org/10.1017/fms.2019.4  

   THURSTON, DYLAN P. (January 2019, Forum of Mathematics, Sigma)

   An elastic graph is a graph with an elasticity associated to each edge. It may be viewed as a network made out of ideal rubber bands. If the rubber bands are stretched on a target space there is an elastic energy . We characterize when a homotopy class of maps from one elastic graph to another is loosening , that is, decreases this elastic energy for all possible targets. This fits into a more general framework of energies for maps between graphs. 

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5. Robust folding of elastic origami

   https://doi.org/10.1039/D2SM00369D  

   Lee-Trimble, M. E.; Kang, Ji-Hwan; Hayward, Ryan C.; Santangelo, Christian D. (August 2022, Soft Matter)

   Self-folding origami, structures that are engineered flat to fold into targeted, three-dimensional shapes, have many potential engineering applications. Though significant effort in recent years has been devoted to designing fold patterns that can achieve a variety of target shapes, recent work has also made clear that many origami structures exhibit multiple folding pathways, with a proliferation of geometric folding pathways as the origami structure becomes complex. The competition between these pathways can lead to structures that are programmed for one shape, yet fold incorrectly. To disentangle the features that lead to misfolding, we introduce a model of self-folding origami that accounts for the finite stretching rigidity of the origami faces and allows the computation of energy landscapes that lead to misfolding. We find that, in addition to the geometrical features of the origami, the finite elasticity of the nearly-flat origami configurations regulates the proliferation of potential misfolded states through a series of saddle-node bifurcations. We apply our model to one of the most common origami motifs, the symmetric “bird's foot,” a single vertex with four folds. We show that though even a small error in programmed fold angles induces metastability in rigid origami, elasticity allows one to tune resilience to misfolding. In a more complex design, the “Randlett flapping bird,” which has thousands of potential competing states, we further show that the number of actual observed minima is strongly determined by the structure's elasticity. In general, we show that elastic origami with both stiffer folds and less bendable faces self-folds better. 

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